Compactness of the immersion $C^1(\bar\Omega)\subseteq C^{\alpha}(\bar\Omega)$

Question

Thanks to all who will answer me! (and sorry for my bad english...)
I have to prove the compactness of the immersion $C^1(\bar\Omega)\subseteq C^{\alpha}(\bar\Omega)$ with (respectively) the usual norms $||f||_{C^1(\bar\Omega)}=||f||_0+\sum_{k=1}^{n}||\partial_k f(x)||_0$ and $||f||_{\alpha}=sup_{x,y\in\bar\Omega| x\neq y}{\frac{|f(x)-f(y)|}{|x-y|}}$.
I know I have to use Ascoli-Arzelà's Theorem so I've tried this method:

1. I've proved the inclusion (I think) because $f\in C^1(\bar\Omega) \implies \forall x,y\in\bar\Omega$ with $x\neq y$ we have $\frac{|f(x)-f(y)|}{|x-y|}\leq\frac{|<\nabla f(z),x-y>|}{|x-y|}\leq\frac{|\nabla f(z)||x-y|}{|x-y|}\leq||f||_{C^1(\bar\Omega)} \implies f\in C^{\alpha}(\bar\Omega)$. Is this ok?
2. Then we can take a sequence $(f_i)_{i\in\mathbb{N}}\subseteq K\subseteq C^{\alpha}(\bar\Omega)$ with $K$ limited. This implies that $\exists M>0$ such that $||f_i||_{\alpha}\leq M$ $\forall i\in\mathbb{N}$. So (I've called the immersion $I$) we have:
   $\cdot$) $|I(f_i)(x)|\leq ||f_i||_{\alpha}\leq M$, so $I(f_i)$ is uniformly bounded.
   $\cdot$) $|I(f_i)(x)-I(f_i)(y)|\leq \frac{|I(f_i)(x)-I(f_i)(y)|}{|x-y|}\cdot |x-y|\leq M\cdot|x-y|$, so $I(f_i)$ is uniformly equicontinuous.
   Can I declare that $I$ is compact?